We report a single-mode diode-pumped, passively mode-locked Cr:LiSAF laser with gigahertz (GHz) repetition rate and kilowatt peak power. A low-loss saturable Bragg reflector with low modulation depth and optimized dispersion compensation mirrors enables the generation of stable, cw mode-locked, sub-100-fs pulses at 1 GHz repetition rate around 865 nm when pumping with four or six laser diodes. Using a 0.25% output coupler and a total absorbed pump power of 0.9 W from six laser diodes, 55 fs, 110 pJ pulses are produced. This corresponds to 1.8 kW of peak power, which to our knowledge represents a record result for GHz Cr:colquiriite lasers.